Quantum signatures of bistability and limit cycle in Kerr-modified cavity magnomechanics

We study a Kerr-modified cavity magnomechanical system with a focus on its bistable regime. We identify a distinct parametric condition under which bistability appears, featuring two stable branches and one unstable branch in the middle. Interestingly, our study reveals a unique transition where the upper branch loses its stability under a sufficiently strong drive, giving rise to limit cycle oscillation. Consequently, we report a rich phase diagram consisting of both bistable and periodic solutions and study quantum correlations around them. While in the bistable regime, we find the entanglement reaching different steady state value, in the unstable regime, entanglement oscillates in time. This study is especially important in understanding quantum entanglement at different stable and unstable points arising in a Kerr-modified cavity magnomechanical system.